Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | C3⋊#85557 |
| Fitting | C3×H(3,𝔽4) ⋊ D3 |
| derived | C3×#5465 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C3⋊#5934 |
| commutator, cosocle, hypocenter | C3×#5465 | C2 |
| Frattini, hypercenter | C22 | C3⋊#1020 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C2×C3 | #5934 |
| LCS | G ▷ C3×#5465 | |
| UCS | C1 ◁ C2 ◁ C22 | |
330 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2,3^2,2^6|b^a=b^2,c^a=c^2,d^a=de,f^a=fg,h^a=hi,d^b=e,e^b=de,f^b=g,g^b=fg,f^d=fi,g^d=ghi,f^e=fh,g^e=gi33 conjugacy classes — view the full character table.
| maximal subgroup of | #72230883, D3×#85557, #72241554 |
| maximal normal subgroup of | D3×#85557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157138);